Find the first term the common difference and give a recursi

Find the first term, the common difference, and give a recursive formula for the arithmetic

sequence. 7th term is 78; 14th term is 57. Show all work please.

Solution

Let the 1^st term of the arithmetic sequence be a and let the common difference be b. Then the nth term of this arithmetic sequence is a + (n -1) b. On substituting n = 7, we get 78 = a + (7-1)b or, a + 6b = 78…(1). Also, on substituting n = 14, we get 57 = a + (14-1)b or, a + 13b = 57…(2). Now, on subtracting the 1^st equation from the 2^nd equation, we get a + 13b – (a + 6b) = 57 – 78 or, 13b – 6b = -21 or, 7b = -21 so that b = -3. On substituting b = -3 in the 1^st equation, we get 78 = a + 6*(-3) or, a – 18 = 78 so that a = 78 + 18= 96. Thus the 1^st term of the given arithmetic sequence is 96 and the common difference is -3. We can verify the result by substituting a = 96 and b = -3 in the 2^nd equation, which gives 96 + 13*(-3) = 57 or, 96- 39 = 57 which is correct. Hence our answer is correct.